2000-2500 USD budget for an integrated amplifier. Do amplifiers sound the same? :)

Low to mid watt tube integrated amp. Choices are many but Primaluna, Rogue and Raven Blackhawk should be on the shortlist.  Just my 2 cents because Klipsch are a relatively simple load.  

I agree with Primaluna integrated wonderful for the price

Eric Squires is right again!

I don’t think all this theory is helping you OP. You seem to be spending effort making an argument.  You may want to try listening to different amps with the Fortes. 

I’ve had the Fortes for nearly a year and I listen about 20 hours a week give it take. I’m not telling you what amp to get or what you’ll like. What I am saying is your speaker sounds different with different amps and does things differently with them. 

I think it is nearly undisputed that the only way to know is to try things out.  I’m no way an expert but one thing I know like audio gospel truth is one must explore in order to discover. Some are talking theories when you need to be talking sound. 

If budget and future proof are a are a top priority I would caution against trying to get this right on the first go. I’d start with a used SS or tube amp in some palatable price—e.g. $750.  

I have several thoughts on this discussion, but one thing stood out for me:

He's got a set of studio monitors but often we need to check our work on this Klipsch Forte III hifi system.

You are saying that you use this system as sort of a "mastering reference' for recording and mixing.  I would keep in mind that the Klipsch speakers are very excellent and sound very "live" and "lively", however the horn drivers are very forgiving and soft in the high frequencies.  This means they are not as clean/clear or as ruthlessly revealing as other speakers.  This may not be what you want if you are "checking the master mix".  You really want to make sure your mix sounds good on an extremely revealing system as well as a budget low end system (at least that's the goal, right?).

That being said, if you switch out the speakers, you may find that the Cambridge will sound rather solid-state and sterile in comparison to what you heard with the Klipsch horn speakers.  So, it may come down to replacing both the speaker and the amp.  It depends on how important the "enjoyment" of the system is to you.  Or if you are just using it as a "tool for master mix check".  Or, you may end up liking the Cambridge just fine with new speakers.  I would also agree that tube amp is not the answer as tubes will generally be even slower and warmer sounding than solid state.  As a listener, the combo with Klipsch may be very nice to listen to, but keep in mind that tubes can also be very colored sounding and will not be as revealing as solid-state.

     I agree a tube integrated would sound the most pleasing to most listeners, including Egoq, but he and his brother will be using his system for monitoring recordings and he stated "so it’d be good for us to have a system with the lowest distortion and flattest response possible". Meeting this goal seems to rule out the use of a tube integrated.
     From Egoq's description, I think class D amplification is likely his best option. From personal experience switching from good class AB amps(McCormack, Adcom and Aragon) to good class D amps(ClassD Audio, Emerald Physics and D-Sonic) in my separates system, I can definitely state with certainty that good class D amps are closer to the amplifier ideal of "a straight wire with gain" than tube or the good class AB amps I’ve previously used. The main characteristics of good class D amps are their very low distortion and very neutral sound, meaning nothing is added to or subtracted from the inputted signals except amplification to drive your speakers; like a straight wire with gain.
     I initially paired my beloved VTL 2.5L preamp with four expensive NOS (new old stock) Mullard tubes with my first class D amp in my system, a ClassD Audio SDS-440CS stereo amp. I was under the impression that the VTL/tubes were needed in my system to add the same warmth, bloom and dimensionality I enjoyed just like when using my VTL with my previous good class AB amps. This worked well with the influence of the VTL/tubes preamp being even more pronounced when paired with the new class D amp, probably due to its ability to just faithfully amplify the inputted signals from the VTL preamp without alteration.
     In an effort to streamline my combination music and ht system about a year later, I removed the VTL preamp and a Parasound AVC2500 5.1 surround sound processor from my system. I used my OPPO-105 Blu-ray player as the heart of this newly streamlined system, connecting all my amps directly to the OPPO’s analog outputs; I connected a newly purchased pair of D-Sonic M3-600-M class D mono-blocks via the OPPO’s XLR L+R stereo outputs, my ClassD Audio stereo amp via the OPPO’s RCA L+R rear surround outputs, a newly purchased Emerald Physics EP100.2EP stereo amp, run in bridged mono mode, via the OPPO’s RCA center channel output and my sub system still connected via the OPPO’s RCA sub output.
     I was expecting ht content, CDs and hi-res music files played back through the solid state OPPO as preamp along with all class D amps to lack the warmth, bloom and dimensionality formerly provided by the now removed VTL/tube preamp. I was pleasantry surprised to discover that most of my familiar mainly acoustic rock and jazz music still possessed the same warmth, bloom and dimensionality on good recordings. However, the "on good recordings" qualifier is important since, with a now obviously high quality but very neutral and revealing solid state preamp section of the OPPO paired with a high quality but very neutral and revealing pair of class D mono-block amps, the quality of all recordings suddenly became clearly obvious, too.
     Everything was now very transparent, accurate and detailed. The warmth, bloom and realistic dimensionality, that I thought my VTL preamp with NOS tubes was providing, was all still there but only if it existed on the music itself AND the recording was good enough to allow these qualities to be evident upon playback on a good, transparent, accurate and detailed system.
     Unfortunately, not all of my favorite and familiar music recordings survived the scrutiny of my currently very transparent system. I found that all of my Coldplay CDs suddenly sounded considerably worse on my home system but still sounded good on my less transparent car system. On the plus side, I can now not only clearly discern the quality of recordings but also any changes made to my system, including interconnect/ speaker cabling and power cords.
     Sorry it’s taken me this long to get to my main point but I thought it was important that you thoroughly understand why I think a class D integrated amp, with a solid state and not a tube preamp section, would most likely be the best option to provide the qualities you desire in your system. You can google ’class D integrated amps’, ’class D integrated amps under $2,500’ or other permutations to see your options.
     A lot of integrated amps now have features you may or not want such as built in streaming, dacs, phono hookup capability, room control etc. I also wanted to point out that going to separates may fit your needs better now or in the future since you’re able to upgrade any specific part of your system more easily because multiple parts are not combined in one box or component. Just something else to consider weighed against the cost of additional interconnects required.

Tim